Nuclear Structure I Single-particle models P Van Isacker

Nuclear Structure (I) Single-particle models P. Van Isacker, GANIL, France NSDD Workshop, Trieste, April-May 2008

Overview of nuclear models • Ab initio methods: Description of nuclei starting from the bare nn & nnn interactions. • Nuclear shell model: Nuclear average potential + (residual) interaction between nucleons. • Mean-field methods: Nuclear average potential with global parametrisation (+ correlations). • Phenomenological models: Specific nuclei or properties with local parametrisation. NSDD Workshop, Trieste, April-May 2008

Nuclear shell model • Many-body quantum mechanical problem: • Independent-particle assumption. Choose V and neglect residual interaction: NSDD Workshop, Trieste, April-May 2008

Independent-particle shell model • Solution for one particle: • Solution for many particles: NSDD Workshop, Trieste, April-May 2008

Independent-particle shell model • Anti-symmetric solution for many particles (Slater determinant): • Example for A=2 particles: NSDD Workshop, Trieste, April-May 2008

Hartree-Fock approximation • Vary i (ie V) to minize the expectation value of H in a Slater determinant: • Application requires choice of H. Many global parametrizations (Skyrme, Gogny, …) have been developed. NSDD Workshop, Trieste, April-May 2008

Poor man’s Hartree-Fock • Choose a simple, analytically solvable V that approximates the microscopic HF potential: • Contains – Harmonic oscillator potential with constant . – Spin-orbit term with strength . – Orbit-orbit term with strength . • Adjust , and to best reproduce HF. NSDD Workshop, Trieste, April-May 2008

Harmonic oscillator solution • Energy eigenvalues of the harmonic oscillator: NSDD Workshop, Trieste, April-May 2008

Energy levels of harmonic oscillator • Typical parameter values: • ‘Magic’ numbers at 2, 8, 20, 28, 50, 82, 126, 184, … NSDD Workshop, Trieste, April-May 2008

Why an orbit-orbit term? • Nuclear mean field is close to Woods-Saxon: • 2 n+l=N degeneracy is lifted El < El-2 < NSDD Workshop, Trieste, April-May 2008

Why a spin-orbit term? • Relativistic origin (ie Dirac equation). • From general invariance principles: • Spin-orbit term is surface peaked diminishes for diffuse potentials. NSDD Workshop, Trieste, April-May 2008

Evidence for shell structure • Evidence for nuclear shell structure from – 2+ in even-even nuclei [Ex, B(E 2)]. – Nucleon-separation energies & nuclear masses. – Nuclear level densities. – Reaction cross sections. • Is nuclear shell structure modified away from the line of stability? NSDD Workshop, Trieste, April-May 2008

Ionisation potential in atoms NSDD Workshop, Trieste, April-May 2008

Neutron separation energies NSDD Workshop, Trieste, April-May 2008

Proton separation energies NSDD Workshop, Trieste, April-May 2008

Liquid-drop mass formula • Binding energy of an atomic nucleus: • For 2149 nuclei (N, Z ≥ 8) in AME 03: av 16, as 18, ac 0. 71, a's 23, ap 6 rms 2. 93 Me. V. C. F. von Weizsäcker, Z. Phys. 96 (1935) 431 H. A. Bethe & R. F. Bacher, Rev. Mod. Phys. 8 (1936) 82 NSDD Workshop, Trieste, April-May 2008

The nuclear mass surface NSDD Workshop, Trieste, April-May 2008

The ‘unfolding’ of the mass surface NSDD Workshop, Trieste, April-May 2008

Modified liquid-drop formula • Add surface, Wigner and ‘shell’ corrections: • For 2149 nuclei (N, Z ≥ 8) in AME 03: av 16, as 18, ac 0. 71, Sv 35, ys 2. 9, ap 5. 5, af 0. 85, aff 0. 016 rms 1. 16 Me. V. NSDD Workshop, Trieste, April-May 2008

Shell-corrected LDM NSDD Workshop, Trieste, April-May 2008

Shell structure from Ex(21) NSDD Workshop, Trieste, April-May 2008

Evidence for IP shell model • Ground-state spins and parities of nuclei: NSDD Workshop, Trieste, April-May 2008

Validity of SM wave functions • Example: Elastic electron scattering on 206 Pb and 205 Tl, differing by a 3 s proton. • Measured ratio agrees with shell-model prediction for 3 s orbit. J. M. Cavedon et al. , Phys. Rev. Lett. 49 (1982) 978 NSDD Workshop, Trieste, April-May 2008

Variable shell structure NSDD Workshop, Trieste, April-May 2008

Beyond Hartree-Fock • Hartree-Fock-Bogoliubov (HFB): Includes pairing correlations in mean-field treatment. • Tamm-Dancoff approximation (TDA): – Ground state: closed-shell HF configuration – Excited states: mixed 1 p-1 h configurations • Random-phase approximation (RPA): Correlations in the ground state by treating it on the same footing as 1 p-1 h excitations. NSDD Workshop, Trieste, April-May 2008

Nuclear shell model • The full shell-model hamiltonian: • Valence nucleons: Neutrons or protons that are in excess of the last, completely filled shell. • Usual approximation: Consider the residual interaction VRI among valence nucleons only. • Sometimes: Include selected core excitations (‘intruder’ states). NSDD Workshop, Trieste, April-May 2008

Residual shell-model interaction • Four approaches: – Effective: Derive from free nn interaction taking account of the nuclear medium. – Empirical: Adjust matrix elements of residual interaction to data. Examples: p, sd and pf shells. – Effective-empirical: Effective interaction with some adjusted (monopole) matrix elements. – Schematic: Assume a simple spatial form and calculate its matrix elements in a harmonicoscillator basis. Example: interaction. NSDD Workshop, Trieste, April-May 2008

Schematic short-range interaction • Delta interaction in harmonic-oscillator basis: • Example of 42 Sc 21 (1 neutron + 1 proton): NSDD Workshop, Trieste, April-May 2008

Large-scale shell model • Large Hilbert spaces: – Diagonalisation : ~109. – Monte Carlo : ~1015. • Example : 8 n + 8 p in pfg 9/2 (56 Ni). M. Honma et al. , Phys. Rev. C 69 (2004) 034335 NSDD Workshop, Trieste, April-May 2008

The three faces of the shell model NSDD Workshop, Trieste, April-May 2008

Racah’s SU(2) pairing model • Assume pairing interaction in a single-j shell: • Spectrum 210 Pb: NSDD Workshop, Trieste, April-May 2008

Solution of the pairing hamiltonian • Analytic solution of pairing hamiltonian for identical nucleons in a single-j shell: • Seniority (number of nucleons not in pairs coupled to J=0) is a good quantum number. • Correlated ground-state solution (cf. BCS). G. Racah, Phys. Rev. 63 (1943) 367 NSDD Workshop, Trieste, April-May 2008

Nuclear superfluidity • Ground states of pairing hamiltonian have the following correlated character: – Even-even nucleus ( =0): – Odd-mass nucleus ( =1): • Nuclear superfluidity leads to – Constant energy of first 2+ in even-even nuclei. – Odd-even staggering in masses. – Smooth variation of two-nucleon separation energies with nucleon number. – Two-particle (2 n or 2 p) transfer enhancement. NSDD Workshop, Trieste, April-May 2008

Two-nucleon separation energies • Two-nucleon separation energies S 2 n: (a) Shell splitting dominates over interaction. (b) Interaction dominates over shell splitting. (c) S 2 n in tin isotopes. NSDD Workshop, Trieste, April-May 2008

Pairing gap in semi-magic nuclei • Even-even nuclei: – Ground state: =0. – First-excited state: =2. – Pairing produces constant energy gap: • Example of Sn isotopes: NSDD Workshop, Trieste, April-May 2008

Elliott’s SU(3) model of rotation • Harmonic oscillator mean field (no spin-orbit) with residual interaction of quadrupole type: J. P. Elliott, Proc. Roy. Soc. A 245 (1958) 128; 562 NSDD Workshop, Trieste, April-May 2008